L-point quadrupole order under magnetic field in cubic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>PrIr</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>Zn</mml:mi> <mml:mn>20</mml:mn> </mml:msub>

We study quadrupole orders in heavy-fermion compound PrIr$_2$Zn$_{20}$ under magnetic fields on the basis of the Landau theory. Assuming $E_g$ electric quadrupolar orders in the cubic symmetry with the ordering wavenumber at the L points in the face-centered cubic lattice Brillouin zone as observed experimentally, we construct a Landau free energy and analyze the resulting free energy. We find that the unidentified high-temperature ordered phase under the magnetic field ${\bf H} \parallel [001]$ reported earlier arises from the rotation of the quadrupole moments of f electrons on the Pr site. We also discuss the phase diagram for other magnetic-field directions and possible double-${\bf q}$ quadrupolar orders in this system.